18776 | Contextual definitions eliminate descriptions from contexts |
Full Idea: A 'contextual' definition shows how to eliminate a description from a context. | |
From: Bernard Linsky (Quantification and Descriptions [2014], 2) | |
A reaction: I'm trying to think of an example, but what I come up with are better described as 'paraphrases' than as 'definitions'. |
21704 | 'Impredictative' definitions fix a class in terms of the greater class to which it belongs |
Full Idea: The ban on 'impredicative' definitions says you can't define a class in terms of a totality to which that class must be seen as belonging. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
A reaction: So that would be defining 'citizen' in terms of the community to which the citizen belongs? If you are asked to define 'community' and 'citizen' together, where do you start? But how else can it be done? Russell's Reducibility aimed to block this. |
21705 | Reducibility says any impredicative function has an appropriate predicative replacement |
Full Idea: The Axiom of Reducibility avoids impredicativity, by asserting that for any predicate of given arguments defined by quantifying over higher-order functions or classes, there is another co-extensive but predicative function of the same type of arguments. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
A reaction: Eventually the axiom seemed too arbitrary, and was dropped. Linsky's book explores it. |
18774 | Definite descriptions, unlike proper names, have a logical structure |
Full Idea: Definite descriptions seem to have a logical structure in a way that proper names do not. | |
From: Bernard Linsky (Quantification and Descriptions [2014], 1.1.1) | |
A reaction: Thus descriptions have implications which plain names do not. |
21727 | Definite descriptions theory eliminates the King of France, but not the Queen of England |
Full Idea: The theory of definite descriptions may eliminate apparent commitment to such entities as the present King of France, but certainly not to the present Queen of England. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7.3) |
21719 | Extensionalism means what is true of a function is true of coextensive functions |
Full Idea: With the principle of extensionality anything true of one propositional functions will be true of every coextensive one. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.3) |
21723 | The task of logicism was to define by logic the concepts 'number', 'successor' and '0' |
Full Idea: The problem for logicism was to find definitions of the primitive notions of Peano's theory, number, successor and 0, in terms of logical notions, so that the postulates could then be derived by logic alone. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7) | |
A reaction: Both Frege and Russell defined numbers as equivalence classes. Successor is easily defined (in various ways) in set theory. An impossible set can exemplify zero. The trouble for logicism is this all relies on sets. |
21721 | Higher types are needed to distinguished intensional phenomena which are coextensive |
Full Idea: The higher types are needed for intensional phenomena, cases where the same class is picked out by distinct propositional functions. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.4) | |
A reaction: I take it that in this way 'x is renate' can be distinguished from 'x is cordate', a task nowadays performed by possible worlds. |
21703 | Types are 'ramified' when there are further differences between the type of quantifier and its range |
Full Idea: The types is 'ramified' because there are further differences between the type of a function defined in terms of a quantifier ranging over other functions and the type of those other functions, despite the functions applying to the same simple type. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
A reaction: Not sure I understand this, but it evidently created difficulties for dealing with actual mathematics, and Ramsey showed how you could manage without the ramifications. |
21714 | The ramified theory subdivides each type, according to the range of the variables |
Full Idea: The original ramified theory of types ...furthern subdivides each of the types of the 'simple' theory according to the range of the bound variables used in the definition of each propositional function. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6) | |
A reaction: For a non-intiate like me it certainly sounds disappointing that such a bold and neat theory because a tangle of complications. Ramsey and Russell in the 1920s seem to have dropped the ramifications. |
21713 | Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? |
Full Idea: It is often thought that Logicism was a failure, because after Frege's contradiction, Russell required obviously nonlogical principles, in order to develop mathematics. The axioms of Reducibility, Infinity and Choice are cited. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6) | |
A reaction: Infinity and Choice remain as axioms of the standard ZFC system of set theory, which is why set theory is always assumed to be 'up to its neck' in ontological commitments. Linsky argues that Russell saw ontology in logic. |
21715 | For those who abandon logicism, standard set theory is a rival option |
Full Idea: ZF set theory is seen as a rival to logicism as a foundational scheme. Set theory is for those who have given up the project of reducing mathematics to logic. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.1) | |
A reaction: Presumably there are other rivals. Set theory has lots of ontological commitments. One could start at the other end, and investigate the basic ontological commitments of arithmetic. I have no idea what those might be. |
21729 | Construct properties as sets of objects, or say an object must be in the set to have the property |
Full Idea: Rather than directly constructing properties as sets of objects and proving neat facts about properties by proxy, we can assert biconditionals, such as that an object has a property if and only if it is in a certain set. | |
From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7.6) | |
A reaction: Linsky is describing Russell's method of logical construction. I'm not clear what is gained by this move, but at least it is a variant of the usual irritating expression of properties as sets of objects. |